Equivalence of vector norms

problem:For each of the following, verify the inequality and give an example of a nonzero vector or matrix for which equality is achieved. In this problem x is an m-vector and A is an matrix.

(a) ,

(b) ,

(c) ,

(d) .

attempt:

(a) I rewrite the inequality so it is easier for me to read:
Let be the largest component of . Then .
If the remaining components are zero, the inequality becomes an equality. If one or more of the remaining components are nonzero, it becomes an inequality.
Example of vector: .

I'm sure there are better ways to verify this.

(b) Again, I rewrite:

.
Let be the largest component of .
Since (m-times) and since for are smaller than , the inequality holds.
It's an equality for all vectors with

I am not at all sure what to do with (c) and (d).
Any hints are appreciated.

Yes. A matrix is a block of nm numbers, so it can be identified with a vector in R^{nm}, and its norm is the same as the norm of this vector, so this makes (a) stronger than (c) I believe. For (d), consider an n by 1 matrix all of whose entries are C>0. Then (d) asserts which is false if n>1, so perhaps there was a typo? If m is replaced by nm then (d) follows from (b) in the same way that (c) follows from (a).

If I look at the matrix A as a -dimensional vector and then calculate the, say 2-norm, I am calculating the Forbenius norm. is the matrix norm induced by vector norms, and so is not the same as the Forbenius norm.
Am I way off here?